Structural optimization of base engine component

Structural optimization ofbase engine componentMaster’s thesis in Applied Mechanics

JOHAN SPARLUND

Department of Applied MechanicsCHALMERS UNIVERSITY OF TECHNOLOGYGothenburg, Sweden 2017

MASTER’S THESIS 2017:36

Structural optimization of base engine component

JOHAN SPARLUND

Department of Applied MechanicsDivision of Material and Computational Mechanics

CHALMERS UNIVERSITY OF TECHNOLOGY

Gothenburg, Sweden 2017

Structural optimization of base engine componentJOHAN SPARLUND

Supervisor: Magnus Levinsson, Volvo Car CorporationExaminer: Magnus Ekh, Department of Applied Mechanics

Master’s Thesis 2017:36Department of Applied MechanicsDivision of Material and Computational MechanicsChalmers University of TechnologySE-412 96 GothenburgTelephone: +46 31 772 1000

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Structural optimization of base engine componentJOHAN SPARLUNDDepartment of Applied MechanicsChalmers University of Technology

AbstractThe purpose of this thesis has been to investigate the potential implementation of structural optimizationsoftware in the product development process at the department of engine development at Volvo CarCorporation. This has been carried out by performing two trial cases of optimization on two differentbase engine components. The models have been created in ANSA and solved with ABAQUS andTOSCA.

The first trial case was of a simpler nature and involved a smaller component inside the bedplate. Itspurpose was mainly to bring knowledge about FE-modelling and optimization, which it fulfilled. Thesecond trial case was more complex, and aimed to find the optimal material distribution on the engineblock’s exterior surface in order to increase the first eigenfrequency of the component. It succeeded inthe sense that it increased the eigenfrequency, but the optimized topology does not contain rib-likestructures to a large extent, as was hoped. Rather, the majority of the element are spread thinly overthe entire surface.

The ultimate conclusion that can be drawn from the results is that it is complicated to implement theuse of structural optimization software on larger base engine components. Optimization is better suitedfor more isolated components of smaller size, with a limited number of load cases that are clearlydefined. This would give the optimization solver a better chance to find an improved structure. It wouldalso limit the need for simplifications and therefore allow for greater confidence in the FE-model andin turn the optimization results. The findings in this thesis does not completely reject the potentialimplementation of structural optimization software in the product development phase, but does showthat structural optimization is not without its flaws and limitations.

Keywords: topology optimization, TOSCA, engine development

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PrefaceThis master’s thesis has been written as the final part of the master’s programme Applied Mechanicsduring the spring of 2017. It has been carried out a the department of engine development at Volvo CarCorporation. I would like to thank my supervisor Magnus Levinsson for all the help and input, myexaminer Magnus Ekh for guidance and all my co-workers for the warm welcome I have received.

Johan Sparlund, Gothenburg 9/6-2017

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Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Structural optimization 32.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Multiple objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Classes of structural optimization . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Topology optimization in TOSCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Material formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Optimizing for distribution of two materials . . . . . . . . . . . . . . . . . . 62.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Solving the optimization problem in TOSCA . . . . . . . . . . . . . . . . . . . . . 7

2.5.1 Optimization quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.1.1 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.1.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.4 Parameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Trial case 1: bearing support inserts 113.1 FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Load cases and boundary conditions . . . . . . . . . . . . . . . . . . . . . . 123.1.3 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.4 Material formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Topology optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Discussion trial case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Trial case 2: engine block ribs 164.1 FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Topology optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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Contents

4.4 Discussion trial case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Conclusions 205.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1 Find a more suitable component to perform optimization on . . . . . . . . . 215.1.2 Verify results from trial case 1 . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.3 Compare results from different optimization software . . . . . . . . . . . . . 21

Bibliography 22

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1Introduction

This chapter describes the background, purpose and limitations of the thesis and the methods used. Itends with describing an outline of the thesis.

1.1 BackgroundAs the car industry sees an increasing demand on fuel economy, while at the same time customers wishto retain performance, engines are downsized. That leads to smaller engines with higher specific powerwhich increases the loads. This is a difficult task for the engineers to tackle, especially when leadtimes are continuously expected to decrease. The currently used design procedure at the department ofengine development is an iterative process where a proposed structure is analyzed using computationalsimulations and redesigned by hand until it meets the criteria. This is a time consuming method thatrequires both a CAE engineer and a design engineer working together. One way to decrease productdevelopment time and the required number of man-hours is to use optimization software, where theiterative design process has been automated. This type of product development aid requires less inputfrom a design engineer and can create multiple design suggestions quickly. Optimization software hassuccessfully been implemented in other departments’ product development process and it is thereforebelieved it could be used for engine development too. Some smaller attempts has been made withinthe team to evaluate its applicability but no one has had the time to properly dive into the subject.

1.2 PurposeThe purpose of this thesis is to investigate the possibilities of applying structural optimization softwarein the development process within the department.

1.3 MethodThe applicability of topology optimization will be tested by performing two trial cases. The first trialcase will be of a simpler kind, to get experience of working with FE-modelling and optimizationsoftware, and the second more complex. The FE-model and optimization model will be created inthe pre-processor ANSA [1]. The optimization software TOSCA [2] will be used in association withthe FE-solver ABAQUS [3]. ABAQUS and TOSCA are both developed by SIMULIA [4] and couldtherefore be assumed to work well together. Post-processing of the FE-results will be done in META[5] and the optimization results in the viewer supplied by the TOSCA software suit.

1.4 LimitationsThe FE-model will not consider any form of temperature dependence and is limited to linear elastic,isotropic material models. Residual stress from manufacturing will not be considered. The influenceof the particular choice of optimization software and FE-solver will not be looked into.

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1. Introduction

1.5 Thesis outlineThe thesis starts with chapter 2 explaining the theoretical background to structural optimization and itsapplication in TOSCA. Chapter 3 & 4 presents and discusses the methodology and results from thetwo trial cases. The final chapter, 5, presents the conclusions that can be drawn from the work andsuggests future work. The last page contains the references.

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2Structural optimization

In this chapter structural optimization and its application in CAE software is presented. It starts with abrief explanation of the general concept of optimization, followed by a description of its underlyingtheory and application in the optimization software TOSCA. For an in depth explanation of the theorybehind structural optimization the reader is directed towards e.g. Christensen and Klarbring [6]. Thereader is expected to be familiar with the basic concept and purpose of the FE method. For an overviewof the FE method the work by Ottosen and Peterson [7] is recommended reading.

In a strictly mathematical approach, the purpose of an optimization scheme is to find the minimumor maximum value of a function for its admissible input values. The application of this concept tothe subject of structural mechanics is referred to as structural optimization. This translates to findingthe optimal design of a structure and in practice this is performed by discretization of a mechanicalproblem to a computational simulation model. In this applied case of optimization the function pertainsto a structure’s mechanical properties and the admissible input values can be interpreted as the limitingoutside factors for the structure.

2.1 TheoryIn Christensen and Klarbring [6] it is stated that the structural optimization problem consist of thefollowing three parts:

• Objective function, f (x,y). Classifies the goodness of the design, by measuring a quantity suchas weight, eigenfrequencies or elastic strain energy.

• Design variable, x. Describes the structures design.• State variable, y. Represents the structural response, for example displacement or effective stress.

The problem can then be formulated as:Minimize f (x,y) w.r.t. x and y

Subjected to

Behavioral constraints on yDesign constraints on xEquilibrium constraints

(2.1)

The behavioral constraints on the state variable y are often written as as a function of g, as g(y)≤0.Design constraints are constraints pertaining to the design variable x. The equilibrium equations in alinear elastic FE problem look like:

K(x)u = F (x)

where K is the stiffness matrix, F the force vector and u the displacement vector corresponding to yin the problem above. Equation (2.1) is referred to as a simultaneous formulation. The equilibrium andoptimization problems are solved simultaneously and the variables x and y are treated as independentof each other. Another way to formulate the problem is the nested formulation, where the equilibriumequation uniquely defines y for a given x. If one treats u as a function of x via the solution of the

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2. Structural optimization

equilibrium equations then we can write the optimization problem as:Minimize f (x,u(x)) w.r.t. xSubjected to g(x,u(x)) ≤ 0

The nested formulation is what is used when solving the problem numerically. To find the solution onetypically need to find the derivatives of f and g, a process referred to as sensitivity analysis.

2.1.1 Multiple objective functionsIt is possible for an optimization problem to contain multiple objective functions. For a problem withN objective functions the problem is formulated as:

Minimize f (f 1(x,u(x)), . . . , fN(x,u(x))) w.r.t. xSubjected to g(x,u(x)) ≤ 0

As the different objective functions can be contradictory to one another and they are all minimizedseparately, there will not exist a distinct optimal solution. One way to solve this issue is to assign aweight to each objective function, making them more or less important to the overall optimizationobjective and therefore end results. A weight wi is assigned to each objective function. The weightsare commonly scaled to be in the range of 0 to 1, and the sum of them are 1:

f =N∑

i=1

fiwi , wi = [0, 1],N∑

i=1

wi = 1

If a specific objective function is believed to be of greater importance for the design of the structure itis assigned a greater weight – and vice versa.

2.1.2 Classes of structural optimizationGenerally speaking, there are three types structural optimization problems. Depending on what isparametrized, it is usually divided into:

• Topology optimization: the design variable x represents connectivity in the domain. It is themost general formulation of a structural optimization problem.

• Sizing optimization: the design variable x represents some type of structural geometry measure,such as cross-sectional area of a truss.

• Shape optimization: the design variable x represents the form of the boundary of the domain.The connectivity of the structure is not changed and no new boundaries are formed.

The focus of this thesis has been on topology optimization.

2.2 Topology optimization in TOSCATOSCA is a commercial software [2] and it can be used for topology optimization to determine theoptimal material distribution in a given design space, to best meet a the objective function and theconstraints. The basis for this is an FE-model that throughout the optimization process will have itstopology altered. The flow chart in figure 2.1 shows the general outline of the iterative process ofsolving the topology optimization problem using TOSCA. Next to the algorithmic flowchart a simpleexample problem is presented: a clamped bar with a point load. The objective function could pertainto the bar’s stiffness and the goal could be to maximize it. The design constraint could be to decreaseits area by some amount.

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First one has to define the design space, i.e. the set of elements making up the area in which theoptimization software is allowed to perform changes in the topology. In the example case, let usassume every element is in the design space. The concept of design space and element’s contributionto the global stiffness matrix is presented in section 2.3.

TOSCA initializes the solution process by guessing a design to meet the criteria. The proposed designis analyzed using the FE method. As TOSCA does not supply its own FE solver, a third party solverhas to be employed. As stated earlier, in this thesis ABAQUS has been used, but several others are alsoapplicable. The results from the FE analysis are used as the input in what can be considered the coreprocess of TOSCA: the mathematical optimization and consequent update of the design. The details ofthe optimization algorithms used in TOSCA are described in section 2.5. If, after the design update thesolution has converged (see section 2.5.2 on convergence), the problem is solved. If not, the newlyproposed design is sent through the process again.

Initialize

FE analysis

Optimization

Update design

Converged?

Finished

No

Yes

Figure 2.1: Flowchart of the optimization process.

2.3 Material formulationIn order to find which design that minimizes or maximizes the objective function, one has to describethe design in the spatial volume mathematically. The simplest approach is to use an integer materialformulation wherein an element exists either as material or void. This is done by letting the designvariable vector x represent the relative densities, ρe. An element having a relative density of 1is contributing to the global stiffness, while a relative density of 0 means that the element is notcontributing. Subscript e denotes individual element number and E0 is the Young’s modulus of a givenmaterial. Ω is the design space and one searches for the optimal subset of material points Ωopt. This isreferred to as the 1-0 problem and is shown below.

Ee = ρeE0, ρe =

1 if e ∈ Ωopt

0 if e 6∈ Ωopt

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For an integer formulation it is not possible to derive the sensitivities used to solve the systemnumerically, as the design variable vector is not continuous. Therefore one switches to a continuousmaterial formulation where elements can have non-integer densities in the range of 0 to 1, makingthe design variable vector continuous. The downside of this is that you have to interpret the stiffnesscontribution of an element having intermediate relative density – basically meaning it is partly voidand partly material. This is done using a material penalization scheme, of which TOSCA offers thefollowing two alternatives:

• SIMP (Solid Isotropic Material with Penalization)• RAMP (Rational Approximation of Material Properties)

Both of the above are described in detail in [8]. They interpolate material stiffness in the followingways:

SIMP: Ee = E0ρpe

RAMP: Ee = E0ρe

1 + p(1− ρe)

SIMP is recommended to be used in case of static load cases, and RAMP for dynamic load cases [9].Figure 2.2 depicts the calculated relative density ρe against how the Young’s modulus of an element ispenalized using the SIMP and RAMP method respectively, for different penalty factors p.

0.5 10

0.5

1

p=1

p=2

p=3

ρe

EeE0

(a) SIMP.

0.5 10

0.5

1

p=0

p=2p=3p=1

ρe

EeE0

(b) RAMP.

Figure 2.2: Penalized element stiffness against relative element density using SIMP and RAMP withdifferent penalty factors p.

For a higher penalty factor the number of elements with intermediate densities decrease, which isdesirable. But a too large penalty factor leads to an increased chance of the optimization algorithmstopping on a local minima rather than the global one [8]. Numerical experiments has proven a goodvalue for the penalty factor p to be around 3 for both schemes [9].

2.3.1 Optimizing for distribution of two materialsIt is possible to optimize for the distribution of two materials, rather than just between material andvoid. This is achieved by limiting the lower end of the range of admissible values for the relativedensity. Figure 2.3 depicts a material interpolation curve using SIMP with a penalty factor of p of2, relating relative density to relative stiffness. The shaded area represents non-admissible elementproperties.

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Typically one knows the Young’s modulus of the two materials to be optimized for. In figure 2.3 arelative stiffness of 1 corresponds to the stiffer of the two materials and Elow the weaker. From Elow

one can determine ρlow to be used as the lower end of the admissible relative density range.

ρlow 10

Elow

1

Relative density

Rel

ativ

est

iffne

ss

SIMP, p=2

Figure 2.3: The material interpolation scheme SIMP with a penalty factor p of 2, with a non-zerolowest value of relative density. The grey area represents the non-admissible element properties.

2.4 MeshTOSCA supports a wide variety of element types in the design space: 2D, 3D, different numberof nodes and elements of different orders. While the element formulation has a great affect on theFE-results, it is of lesser importance for the optimization solver itself [9]. The number of elements isstill important though, as a finer mesh will yield a higher resolution structure than a course mesh will.

2.5 Solving the optimization problem in TOSCATOSCA has two distinctly different algorithms for solving the optimization problem, these are referredto as the sensitivity based solver and the controller based solver. Both have their pros and cons, andwhich one to use depends on the type of optimization problem at hand. The controller based solver ismore limited in terms of quantities that can be used, and one often founds that more complex sensitivitybased solver is needed. Below follows a brief description of the two:

Sensitivity based solverThe sensitivity based solver uses the Method of Moving Asymptotes (MMA), described in [10]. Thesensitivity based approach offers a wider variety of possible constraints and objective functions than itscounterpart. The complete list of available design responses to use can be found in [9], but for exampleit could be: eigenfrequencies, reaction forces or center of gravity. The solver uses semi-analyticalsensitivities based on finite differences to update the stiffness matrix:

∂K

∂x=

K0+p −K0

∆x

K is the updated stiffness matrix pertaining to the updated design and K0 is the stiffness matrix forthe unchanged current state. K0+p is the perturbed stiffness matrix, where the design parameters hasbeen changed slightly. The denominator ∆x is the change in the design variable values. To find thesensitivities TOSCA linearises around the current state by adding so-called matrix steps and pseudoloads to the ABAQUS FE-model. While these steps have to be solved for, they do not affect theFE-results and are only needed in the iteration process to solve the optimization problem. Dependingon what quantity is optimized for, TOSCA adds a different amount of matrix steps and pseudoloads. Compliance (a quantity explained in section 2.5.1) and eigenfrequency are good quantities to

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optimize for, partially because it does not add pseudo loads, unlike what for example a minimizationof displacement would.

Controller based solverThe controller based solver does not solve for the sensitivities, instead the strain energy and gridpoint stresses are used. It is limited to constraints on volume and an objective function pertaining tocompliance. The TOSCA manual [9] does not go into any details in explaining the inner workings ofthe solver, why there are uncertainties regarding its function. The advantage of the controller basedapproach is mainly that it is faster than the sensitivity based.

2.5.1 Optimization quantitiesWhat quantities to constrain and optimize for depends on the goal of the project and the limitations ofthe FE-model and optimization solver. The choice of objective function measurement and constraintsare crucial as they are the main deciding factors of the end results.

2.5.1.1 Objective function

The following quantities has been used as objective functions in this thesis:

ComplianceCompliance is quantity commonly used in optimization, that is seldom seen elsewhere. It is a scalarquantity that measure the elastic strain energy. It is usually denoted C, and it inversely relates to thestructures stiffness. Typically one wants to minimize the compliance, as a lower compliance equals ahigher stiffness. It is formulated as:

C = F (x)Tu

EigenfrequenciesA structural optimization model involving eigenfrequencies is less straight forward to define than oneusing compliance. Typically one wishes to include the first or the first few eigenmodes, which leads toa multi-objective function problem. It is recommended to use the supplied Kreisselmaier-Steinhausermethod for weighing the objective functions [9]. The objective function is then defined as:

fKS = −1

kln(

N∑i=1

exp(kfi)) , by default k =30

fmin

k is the weight for each eigenmode (see section 2.1.1 for concept of weighted objective function), fi

are the eigenfrequencies and N is the number of eigenmodes included in the objective function. Thismethod eliminates the need for mode tracking which is computationally expensive.

2.5.1.2 Constraints

The following constraints have been used in different capacities in this thesis:

SymmetryA design space can be constrained to various forms of symmetry. Note, this does not require asymmetric mesh in TOSCA.

VolumeVolumetric constrain on a design space is possibly the most straight forward to understand. The designspace is constrained to decrease its volume by a specific amount, as to limit extent of the materialusage.

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2.5.2 ConvergenceTOSCA determines convergence based on the change in value in either the objective function or therelative densities. If the current design iteration fulfills the constraints and the percentage wise changein the convergence measurement from the previous iteration is deemed small enough, the solution issaid to have converged. Convergence in objective function or relative densities does not necessarilytranslate to a converged topology, therefore it can be beneficial to force the optimization algorithm toperform at least a certain numbers of iterations.

As there is nothing that says that a converged solution necessarily has reached the global optimumthe solution will probably stop on what is one of several local optima. This means that it possiblyexists multiple solutions that could have been reached, depending on for example the starting solutionguess. Therefore it can be a good idea to run multiple versions of the same optimization problem withdifferent starting guesses on the relative element densities, to confirm that they reach a similar solution.

2.5.3 FilteringIt is necessary to process the optimization solver’s results in order to limit mesh dependency and theexistence of numerical instabilities, a process referred to as filtering. A common numerical issueresolved by filtering is the occurrence of checkerboard patterns. It refers to neighbouring elementsalternating between being material and void in a checkerboard-like pattern. This type of material layoutmakes sense from a mathematical perspective, but not in a mechanical or manufacturing sense. As theelements are only in contact in their infinitesimal corners, they can not carry loads. This numericalissue is resolved with what is referred to as a sensitivity filter, developed by Sigmund [11]. It worksby applying a smoothing to the sensitivity analysis. Although its effect on mesh independence andcheckerboard patterns has not been proven mathematically, experience has shown it to be useful. In [8]it is expressed as:

∂f

∂ρe=

1

ρe

N∑i=1Pi

N∑i=1

Piρi∂f

∂ρi, with Pi = rmin − dist(e,i)

N is the total number of elements in the domain. The function dist(e, i) is defined as the distancebetween the center of the current element e and the center of element i, see figure 2.4. Only the shadedelement in said figure have an affect on the filter and consequently the minimum structure size. Thedots are the element center points used by the filter.

rmindist(e,i)

ei

Figure 2.4: An example domain with 49 elements, to demonstrate the sensitivity filter.

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TOSCA offers two sensitivity filters referred to as a STANDARD and LOW, where STANDARD isrecommended for finer meshes and vice versa. What length of filter radius rmin to use is based onexperience and the type of task at hand, but an rmin twice as large as the mean element edge length isrecommended as a starting point [9]. If the filter radius is too large the structure will not be detailed. Atoo small radius will results in an overly fine structure.

2.5.4 Parameter studyIn the previous sections the methods and parameters that make up a structural optimization problemhave been presented. With these, the user will have to make choices that affects the end results. It canbe beneficial to perform small scale parameter study, i.e. to run several similar optimization jobs withvarying settings and various types of constraints. For example one can achieve different results if theeigenfrequencies are to be maximized or simply constrained to be above a specific limit.

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3Trial case 1: bearing support inserts

This chapter presents the methodology and results of performing a topology optimization aiming tofind the ideal material distribution in a part cast into the bedplate. After an introduction of the subjectmatter of the trial case, the FE- and optimization model are explained and the results are presented anddiscussed.

Figure 3.1 shows the engine block and the bedplate. The block and bedplate are connected by someM8 and M10 bolts, not visible in the figure. When the bedplate is cast, four bearing support structuresare cast into it, referred to as inserts. Their purpose is mainly to decrease the risk of fatigue of thebedplate caused by the loads from the crank shaft, that goes through the engine block and the bedplate.The engine block and bedplate are made out of aluminium and the inserts out of the heavier materialcast iron. Due to the heavy weight of cast iron it would be desirable to find the optimal distribution ofcast iron and aluminium in the volume taken up by the insert.

The CAD-models depicted in this chapter are not the actual geometries used in the optimization model.This is because the used CAD-models are still under development and are therefore not suitable forpublishing in a master’s thesis. The components depicted are from an older generation, but they stillhold the same functions and the geometries are similar. Note that the model colors are not related tomaterial properties, and are chosen only as to more easily distinguish different components.

Block

Bedplate

Cylinder lining

Figure 3.1: Isometric view of the block, bedplate and cylinder lining. Not visible are the main bearinginserts and the bolts.

3.1 FE-modelGiven the iterative nature of solving the optimization problem it is a time consuming process, why it iscrucial to not have too computationally expensive FE-model. In order to avoid this, simplificationsoften have to be made. In this case it means to instead of creating a model representing the entire

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3. Trial case 1: bearing support inserts

engine block it is represented by the two opposing halves of the middle cylinders, see figure 3.2 andcompare with figure 3.1. This reduces the simulation time greatly.

(a) Isometric view of thesimplified model with twoopposing cylinder halves.

Insert

M8

M10

(b) Front view of outlines the simplifiedmodel. Block and bedplate drawn as out-lines, and cylinder liner not visible.

Figure 3.2: Simplified FE-model.

3.1.1 MeshThe mesh was created using the pre-processor ANSA and consists of roughly 1.2 million elements.The insert, bedplate and bottom third of the block have an average node spacing of 1.5mm, while therest of the model has an average of 5mm. The purpose of this is to have a finer discretization wherethe majority of the deformation will happen. The volumetric mesh for all components, except forthe insert, is created with a growth factor of 20% inwards from the surface. The insert has a uniformvolume element size to not skew the optimization results, as larger elements would be favored inthe optimization process. The element type used is a first order (linear) tetrahedral element with 4nodes and 1 integration point, referred to as C3D4 in the ABAQUS documentation [12]. It is a factthat using first order elements will yield an overly stiff response compared to the more accurate andmore computationally heavy second order elements [7]. But it was found by a colleague that whendecreasing the element order from second to first on an engine block FE-model regularly used withinthe department, the results changed only slightly. Based on that, first order elements were consideredaccurate enough to be used for this application.

3.1.2 Load cases and boundary conditionsForces are applied on all the nodes on the inner boundary of the bearing. The load cases are sampled atdifferent crank angles and represent the forces and moments acting on the bearing by the crank shaft.Their magnitude and directions have been obtained by another team within the same department. Thenumber of load cases to include in the simulation is a balance act, as too many would give a too longcomputational time and too few would not accurately represent reality. Four load cases were included,leading to that the FE-problem took about 45 minutes to solve on the computer cluster. All formsof temperature loads are disregarded as they are believed to have little effect on the outcome of theoptimization of the bearing support.

To avoid rigid body motion, the nodes on the top of the engine block are fixed. Before any loadis applied, the bolts connecting the bedplate to the engine block are pre-tensioned. This is done inthree steps: first a specific displacement is prescribed, then a force corresponding to restoring thatdisplacement and finally a step locking the pre-tension.

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3.1.3 ContactThe FE-model has contact conditions in the following locations:

1. Bedplate to bearing support2. Block to two M8 bolts3. Block to two M10 bolts4. Block to cylinder liners5. Block and bedplate to bearing6. Bedplate to engine block7. Bedplate to two M8 bolts8. Bedplate to two M10 bolts

FE problems with contact conditions are non linear and hence computationally heavy. Item 1-4 in thelist above have a contact condition referred to as tie, which ’glues’ the connecting surfaces togetherand essentially makes them one part. This is a simplified form on contact condition used to decreasesimulation time. The downside is that it can result in higher stresses. Item 5 and 6 use a linear penaltymethod with constant friction coefficient. As the bearing is press-fit between the bedplate and theengine block, it will have residual stresses and strain. Item 7 and 8 use the standard ABAQUS surfaceinteraction behaviour (described in [12]), with a constant friction coefficient.

3.1.4 Material formulationThe FE-model consists of four materials:

• Ductile iron (bearing supports)• Aluminium (engine block and bedplate)• Steel (bolts)• GOE (cylinder lining)

All materials are modeled as isotropic and linear elastic. Their only differences lies in the respectiveYoung’s modulus and Poisson’s ratios. Plastic behaviour has been disregarded to reduce simulationtime. Material properties are taken from a state of 20°C and are constant as no thermal analysis isperformed.

3.2 Topology optimization modelThe following section describes the topology optimization model created and the results from it. TheTOSCA-model was set up using ANSA and solved on the VCC computer cluster. The problem isdelimited as follows:

Objective functionMinimizing the compliance of the entire model, which is the equivalent of maximizing its stiffness [9].

Design spaceAll elements making up the main bearing support insert, except for those whose faces are in contactwith the bolts or facing towards the bearing.

ConstraintsThe material distribution has to be symmetric and the design has to decrease its volume by 3%. As thedensity of cast iron is roughly 2.6 times higher than the density of aluminum a volume decrease of 3%corresponds to a weight decrease of just over 1%.

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Solver settingsThe sensitivity based algorithm is used together with SIMP, see section 2.5 and 2.3 respectively. Theprocess is forced to performed 40 iterations. The lower end of the range for the relative density is 0.75as that corresponds to the density relation between aluminum and cast iron, see section 2.3.1.

3.3 ResultsFigure 3.3 shows the optimization results from trial case 1. It can be seen that the elements of thedesign space contributing the least towards the FE-models compliance can be found on the bottom ofthe insert. The elements having a relative density of 0.75 corresponds to aluminum, and the elementshaving a relative density of 1 to cast iron. Due to the symmetry constraints the front view is enough toget a complete picture of the material distribution.

0.75 0.78 0.8 0.83 0.85 0.88 0.9 0.93 0.95 0.98 1

Relative density

Figure 3.3: Optimization results trial case 1. An element having a relative density of 0.75 correspondsto aluminum, and an element having a relative density of 1 to cast iron.

3.3.1 ConvergenceFigure 3.4 shows the normalized objective function convergence and the normalized volume constrainconvergence. The objective function is normalized such that a value of −1 equals the initial complianceof the entire model. The optimized topology reaches almost the same compliance as the original, it isbasically within computational the margin of error. The volume constraint is normalized such that avalue of 1 equals a fulfilled constraint. These figures confirm the objective functions convergence andthat the volume constraint is fulfilled. This together with a visual inspection of change in topologybetween iterations, which is difficult to illustrate in a report, leads to the conclusion of good overallconvergence.

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0 5 10 15 20 25 30 35 40−1

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Vol. constraint fulfilled

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0 5 10 15 20 25 30 35 40

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Figure 3.4: Objective function convergence and volume constraint.

3.4 Discussion trial case 1One of the purposes of the first trial case was to get experience of working with topology optimizationand FE-modelling. Much time was put into learning to operate ANSA and understanding ABAQUSand TOSCA.

From figure 3.4 it is clear the models compliance, i.e. the inverse of the stiffness, is sustained while thevolume constraint is still fulfilled. This would indicate a potential redesign could be made based onthe results. Naturally, there would be some question marks regarding the results. It is possible thatit is overly simplistic to only include two cylinder halves, and that the whole engine block needs tobe included. Furthermore, the load cases included in the FE-model does not describe all the loadsexperienced by the inserts and would therefore not be optimized for real life behaviour. But still, it canbe seen as an indicator that the inserts could stand to lower its weight.

In the case of an objective function pertaining to maximizing stiffness while optimizing for thedistribution of two materials one has to constrain the volume to decrease. If not, the entire design spacewould be occupied by the stiffer material. The constraint of the volume being forced to decrease byspecifically 3% is basically arbitrary. The volume decrease serves as an indicator of which elementscontribute the least towards the structures stiffness. It can be seen as a starting point to remove materialif a redesign of the component were to be done.

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4Trial case 2: engine block ribs

This chapter presents the methodology and results of performing a topology optimization aiming tofind the ideal material distribution on the exterior surface on the long sides of the engine block. After abrief introduction of the trial case, the FE- and optimization models are explained and the results arepresented and discussed.

From a structural point of view, the purpose of the ribs on the exterior surface of the engine block is toincrease its stiffness in order to minimize deformation during loading and to maximize the componentseigenfrequencies. The placement of these is done by design engineers, without the use of optimizationsoftware. Most of them are lined up either strictly vertically or horizontally, which is not the typeof design one would expect to come out of an optimization software. Basically, the approach in thistrial case is to remove rib structures and to replace them with a design space and let the optimizationsoftware find the mathematically optimal design.

The engine block used in this trial case is the same as the one used in chapter 3. But once again theCAD-models depicted are not the actual geometries used in the optimization model, but rather they arefrom an older generation of VCC engines, for the same reason as in the previous chapter.

4.1 FE-modelFigure 4.1(a) depicts on of the long sides of the CAD-model representing the original engine block. Foran isometric view of the engine block see figure 3.1 on page 11. The process to create the FE-model isto first delete the faces making up the rib structures and to replace them with faces aimed to createsmooth and flat surfaces. This was only done on the long sides of the engine block, and the short sideswere left as is. For complex or curved surfaces this can be a difficult and time consuming process.The overall removal and replacement of the ribs was successful, but in a few smaller surface regionspreviously curved surfaces are now plane. This is believed to have limited effect on the end results, asthe eigenmodes were confirmed to be of similar nature when compared in the post-processor META.The engine block without the rib structures is not shown in the report as that work was done on anengine block whose design is still classified.

Figure 4.1(b) depicts the modified engine block without the rib structures, with the added design spaceson each side. The design spaces is created by making an empty box starting at the exterior surface andstretching some distance into the void, perpendicularly.

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4. Trial case 2: engine block ribs

(a) (b)

Figure 4.1: Side view of the unmodified engine block, and the modified ribless engine block with theadded design spaces.

The eigenfrequency used in the optimization model are solved for in ABAQUS using the Lanczosmethod. It is linearized around an unconstrained state. The only material used is aluminum and it isrepresented by an isotropic, linear elastic material model.

4.1.1 MeshThe model consists of three volumes: the engine block itself and the two boxes making up the designspace on both long sides of it. The mesh was created in ANSA and consists of roughly 4 millionelement all together. The element type is the same as in trial case 1, a first order (linear) tetrahedralelement with 4 nodes and 1 integration point. The exterior surface of the engine block where the designspaces starts has a node spacing of 4 millimeters. Similarly as in trial case 1 the volumetric mesh has agrowth rate of 20% inwards for non-design areas, as to decrease the total number of elements and saveon computational time. This is believed to not affect the results in a noteworthy way.

4.2 Topology optimization modelThe topology optimization model was created in ANSA and is set up in the following way:

Objective functionMaximize the first eigenfrequency of the engine block. As the Kreisselmaier-Steinhauser method isused, mode tracking is automatically applied.

ConstraintsThe design space is constrained to decrease its volume on each subdomain to a level that correspondsto the weight difference between the original and modified engine block for that side.

Design spaceSee figure 4.1.

Solver settingsThe sensitivity based algorithm is used together with SIMP, see section 2.5 and 2.3 respectively.Prescribed to perform 100 optimization iterations.

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4.3 ResultsThe resulting topology after the optimization with TOSCA can not be shown in the report due to thefact that the engine block design is still under development. However, the optimization model does notseem to favor the creation of rib-like structures. The majority of the elements were spread thinly andevenly over almost the entire side surfaces, with only small tendencies of creating rib-like structures.This is not in line with what was expected and hoped for.

Table 4.1 shows the resulting eigenfrequency values from the modified ribless topology and theresulting optimized topology. Results are in comparison to the original topology’s first eigenfrequency.It makes sense that the eigenfrequency for the modified engine block is lower than for the originalblock as weight is removed. The 5% increase of the first eigenfrequency for the optimized topologyis obviously good, but as described earlier the material is not distributed in a way that resembles ribstructures. This ultimately renders the results not usable for redesigning the rib structures.

First eigenfrequencyModified ribless topology -2%

Optimized topology +5%

Table 4.1: Comparison of the eigenfrequencies for modified ribless engine block and the optimizedengine block to the original design. Note that the optimized topology is heavier than the modifiedtopology.

4.3.1 ConvergenceFigure 4.2 displays the convergence for objective function and the volume constraints. The objectivefunction is normalized against the eigenfrequency for the original engine block. The volume constraintis normalized such that a value of 1 equals a fulfilled constraint. Conclusion of good convergence isdrawn based on the same argumentation made in section 3.3.1 for trial case 1. Compared to trial case1 the design space geometry is more complex and the number of elements higher. Therefore it is nosurprise that the number of iterations needed to reach convergence is greater.

0 10 20 30 40 50 60 70 80 90 1001

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0 10 20 30 40 50 60 70 80 90 1000.75

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Design space 1Design space 2

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Figure 4.2: Convergence of objective function and volume constraints, normalized against results fromthe engine block model with the added design spaces.

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4.4 Discussion trial case 2As stated previously, the optimization model does not seem to favor the creation of rib structures butrather it wants to smear out the elements over the entire surface. Many different settings has been triedto create a model more favorable towards the creation of rib structures, such as:

• Smaller and larger radius for the checkerboard filter• Different material interpolation scheme settings• Higher and lower mesh densities• The including of manufacturing constraints

But regardless of setting tweaks, the results were of a similar nature of not creating rib-like structures.It would have been interesting to investigate the possibilities of letting the starting guess be the originaldesign and let TOSCA perform changes based on that. This angle of attacking the problem was notinitially considered, as the idea was to let TOSCA create something new and different from the originaltopology.

It is not overly simplistic or erroneous in itself to consider the eigenfrequencies as objective functionsand to constrain the volume. But the eigenfrequency is solved for from a linearization around anunconstrained state, meaning the model is undeformed and free to move in any direction, whichobviously is not a physically correct representation. The upside of such a simple model which onlydoes modal analysis is that it is so fast to solve for that even a model consisting of millions of elementscan perform upwards of 100 optimization iterations on VCC’s computer cluster within a day’s time.The current load case scenario means that any topology results that would be realized to a CAD-modeland used in for example an engine block fatigue simulation model, would be subjected to loads andboundary conditions that it was not optimized against. It would have been interesting to create a morecomplete model in terms of load cases and components, but that gets complicated for multiple reasons:

• Complex to accurately describe all load cases experienced by the engine block• Computationally heavy to include multiple components and many load cases

The load cases experienced by the engine block is a mix of forces originating from the internalcombustion and the effect of having components hanging on the engine. The forces from the internalcombustion acting on the engine block via the crank shaft are relatively easy to describe. But if asufficient amount of those loads and the multiple parts making up the entire engine (block, bedplate,inserts, bolts, bearings, etc.) are included in the FE-model it quickly becomes computationally heavyto solve for. The problem with accounting for components hanging on the side of the engine block isnot necessarily a computational issue as they could probably be described by point loads. However, itwould be difficult to find accurate representations of magnitude and direction of such loads.

It is surprising that the objective function convergence in figure 4.2(a) is not monotonous. Especiallyconsidering that the optimization solver uses a gradient method [10]. This has been discussed with theTOSCA support team and it was assured that it is not a concern, as long as the convergence shows amonotonous trend.

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5Conclusions

This chapter presents the conclusions that can be drawn from the thesis work and outline suggestionsof future work.

The results from trial case 1 could be used as a basis for a future redesign of the main bearing inserts.The compliance measurement indicates the stiffness would be the same even when the weight wasreduced. But as discussed in section 3.4 there is some hesitation towards a realization as the modelwas quite simplified.

The optimization in trial case 2 was successful in the sense that it managed to increase the firsteigenfrequency of the engine block, but unsuccessful in the sense that it did not yield the rib structuresthat were expected and hoped for. But even if the optimization model would have found rib structuresthat increased the first eigenfrequency, there would still be hesitation towards using the new topologyin a design realization. The proposed design would be based on an optimization model not accuratelydescribing all load cases and boundary conditions experienced by the engine block, as explained insection 4.4. The main conclusion that can be drawn from trial case 2 is that optimization with TOSCAworks better on a more isolated component of smaller size, with a limited number of load cases that areclearly defined. This would give less need for simplifications and therefore allow for greater confidencein the FE-model and in turn the optimization results.

The work was set out with the purpose of investigating the possibilities and potential problems ofimplementing the use of structural optimization software in the product development phase within thedepartment. TOSCA certainly can be used to generate useful new ideas and concepts, but there is noguarantee of return on investment in terms of useful results, as was seen in trial case 2. While the basicapplication of TOSCA is relatively easy to perform, there exists a myriad of different settings whichinfluences the end results greatly. An engineer using TOSCA would have to have an understanding ofthese parameters and how they affect the results. That type of knowledge is not necessarily a subset ofthe traditional knowledge of a CAE engineer working primarily with FE analysis on solids. To avoidan overly time consuming learning curve any future endeavours into structural optimization within theteam should to some degree be assisted by an optimization expert.

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5. Conclusions

5.1 Future workThree specific areas of interest for future work have been found and these are presented below.

5.1.1 Find a more suitable component to perform optimization onThe results from the trial case 2 does not support that further work is put into that specific type ofapplications. That does not necessarily rule out the overall applicability of optimization on othercomponents relevant for the the department. Guidelines for choosing components to work on, basedon the experience from this thesis, would be:

• Simple enough FE-model to be able to perform up to 100 optimization iterations within areasonable time frame

• Large enough design space to leave room for the optimization solver to "do its magic"• Comprehensive enough load cases and boundary conditions to give confidence in the results

No consideration has been put into what such a component might be.

5.1.2 Verify results from trial case 1It would be interesting to realize the optimization results from trial case 1 to a CAD-model and to finda way of comparing the optimized design to the original.

5.1.3 Compare results from different optimization softwareThe decision of using TOSCA and ABAQUS rather than the optimization software OptiStruct (whichis also available at VCC) was made without much research about pros and cons of OptiStruct. It wouldbe interesting to compare the softwares and the results from them.

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Bibliography

[1] ANSA. https://www.beta-cae.com/ansa.htm. Accessed: 2017-04-24.[2] TOSCA. https://www.3ds.com/products-services/simulia/products/tosca/. Accessed: 2017-05-27.[3] ABAQUS FEA. https://www.3ds.com/products-services/simulia/products/abaqus/. Accessed:

2017-04-24.[4] SIMULIA 3DS. https://www.3ds.com/products-services/simulia. Accessed: 2017-04-04.[5] META. https://www.beta-cae.com/meta.htm. Accessed: 2017-04-24.[6] Peter W. Christensen and Anders Klarbring. An Introduction to Structural Optimization. Springer,

2008.[7] Niels Saabye Ottosen and Hans Petersson. Introduction to the Finite Element Method. Prentice

Hall, 1992.[8] Martin Philip Bendsoe and Ole Sigmund. Topology Optimization: Theory, Methods, and Applica-

tions. Springer, 2013.[9] FE-DESIGN GmbH. SIMULIA Tosca Structure Documentation 8.1. SIMULIA, 2014.

[10] Krister Svanberg. The method of moving asymptotes - a new method for structural optimization.Simulation, 1987.

[11] Ole Sigmund. On the design of compliant mechanisms using topology optimization. Mechanicsof Structures and Machines, 25(4):493–524, 1997.

[12] 3DS Simulia. Abaqus Online Documentation: Analysis User’s Guide. Dassault Systemes AB,2016.

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Structural optimization of base engine component

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